=Paper=
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|storemode=property
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|pdfUrl=https://ceur-ws.org/Vol-972/invited2.pdf
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<pdf width="1500px">https://ceur-ws.org/Vol-972/invited2.pdf</pdf>
<pre>
                 Multi-adjoint concept lattices

                                     Jesús Medina

                   Dept. Mathematics, University of Cádiz, Spain⋆
                              jesus.medina@uca.es

Abstract. Multi-adjoint concept lattices were introduced [2,3] as a new general ap-
proach to formal concept analysis, in which the philosophy of the multi-adjoint para-
digm [1,4] to formal concept analysis is applied. With the idea of providing a general
framework in which diﬀerent approaches could be conveniently accommodated, the
authors worked in a general non-commutative environment; and this naturally lead
to the consideration of adjoint triples, also called implication triples or bi-residuated
structure as the main building blocks of a multi-adjoint concept lattice.
In recent years there has been an increased interest in studying formal concept analysis
on the perspective of using non-commutative conjunctors. This is not a mere mathe-
matical generalization, but a real need since, for instance, when one learns a conjunction
from examples it is not unusual that the resulting conjunction does not satisfy com-
mutativity. Diﬀerent authors have argued in favour of considering non-commutative
conjunctors. Actually, there exist quite reasonable examples of non-commutative and
even non-associative conjunctors deﬁned on a regular partition of the unit interval.
Hence, the possibility of considering non-commutative conjunctors provides more ﬂex-
ibility and increases the number of applications.
This is one of the properties that the multi-adjoint concept lattice framework oﬀers.
Another important feature is that diﬀerent preferences among the set of attributes
or/and objects can be considered.
Moreover, this framework has been extended following diﬀerent lines and has been
applied to deﬁne general extensions of fuzzy rough sets theory, to solve fuzzy relation
equations, etc.


References
1. P. Julian, G. Moreno, and J. Penabad. On fuzzy unfolding: A multi-adjoint ap-
   proach. Fuzzy Sets and Systems, 154(1):16–33, 2005.
2. J. Medina, M. Ojeda-Aciego, and J. Ruiz-Calviño. Formal concept analysis via
   multi-adjoint concept lattices. Fuzzy Sets and Systems, 160(2):130–144, 2009.
3. J. Medina, M. Ojeda-Aciego, and J. Ruiz-Calviño. On multi-adjoint concept lat-
   tices: deﬁnition and representation theorem. Lecture Notes in Artificial Intelligence,
   4390:197–209, 2007.
4. J. Medina, M. Ojeda-Aciego, and P. Vojtáš. Similarity-based uniﬁcation: a multi-
   adjoint approach. Fuzzy Sets and Systems, 146:43–62, 2004.


⋆
    Partially supported by Spanish Ministry of Science and FEDER funds through
    project TIN09-14562-C05-03 and Junta de Andaluca project P09-FQM-5233.

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